求解Euler方程的高精度熵相容格式研究

发布时间：2018-03-17 12:11

本文选题：Euler　切入点：方程　出处：《长安大学》2015年硕士论文　论文类型：学位论文

【摘要】：双曲守恒律方程组是计算流体力学中的重要研究内容之一。求解双曲守恒律方程组的熵相容格式从热力学第二定律出发,是一种满足熵稳定条件的数值格式,也是目前对熵的变化估计得最准确的一种熵稳定格式。该格式有较强的物理背景,能有效避免一些非物理现象的产生。本文在Euler方程的熵相容格式的基础上,研究了一类高分辨率、高精度的熵相容格式。主要研究内容包括:(1)介绍熵相容格式。论述了熵守恒/熵稳定格式的基本理论、一般熵守恒格式的构造过程,并介绍了两种简单且便于实现的熵守恒格式;然后给出了熵相容格式的具体形式,并引入限制器使数值格式具有高分辨率的性质;最后针对一维Euler方程,将上述格式应用于几个数值算例,验证各种格式的特性。(2)详细论述WENO重构的过程。首先在一维标量方程的情形下给出了WENO重构的过程;然后针对一维Euler方程,给出了通过对守恒型变量进行局部特征变量分解来保证格式的基本无震荡特性的方法,并论述了进行局部特征分解的必要性;最后针对特征分解过程中向量内积运算量较大的问题,引入了一类用压强和熵来代替重构权重计算中的特征变量的方法,以降低计算量。(3)构造高精度的熵相容格式。首先通过不同模板上的熵守恒格式的线性组合得到高阶熵守恒格式;然后将上述WENO重构过程应用于熵相容格式的数值粘性项,与高阶熵守恒格式结合得到高精度的熵相容格式;最后在几个数值算例上验证了格式的高精度特性,且通过压强和熵来构造权重的方法比直接用特征变量构造权重更省时。(4)将熵相容格式、高分辨率熵相容格式推广至二维情形,通过二维Euler方程的几个数值算例验证格式在二维情况下的特性。其中重点研究了圆柱绕流问题。数值算例表明熵相容格式比Roe格式能更好地避免高马赫圆柱绕流问题中的红斑现象。
[Abstract]:Hyperbolic conservation law equations are one of the important research contents in computational fluid mechanics. The entropy compatible scheme for solving hyperbolic conservation law equations is a numerical scheme which satisfies the condition of entropy stability based on the second law of thermodynamics. This scheme has strong physical background and can effectively avoid some non-physical phenomena. This paper is based on the entropy compatible scheme of Euler equation. In this paper, a class of entropy compatible schemes with high resolution and high precision are studied. The main contents of this paper include the introduction of entropy compatible schemes. The basic theory of entropy conservation / entropy stability schemes and the construction process of general entropy conservation schemes are discussed. Two simple and easy to implement entropy conservation schemes are introduced, and then the specific form of entropy compatible scheme is given, and a limiter is introduced to make the numerical scheme have high resolution. Finally, for the one-dimensional Euler equation, The process of WENO reconstruction is discussed in detail by applying the above scheme to several numerical examples to verify the characteristics of various schemes. Firstly, the process of WENO reconstruction is given in the case of one-dimensional scalar equation, and then the one-dimensional Euler equation is discussed. This paper gives a method to guarantee the basic non-oscillatory characteristic of the scheme by decomposing the local characteristic variable of the conserved variable, and discusses the necessity of the local characteristic decomposition. Finally, in order to solve the problem of large computation of vector inner product in the process of feature decomposition, a new method is introduced, which uses pressure and entropy instead of reconstructing the feature variables in weight calculation. The high order entropy conserved scheme is obtained by linear combination of entropy conservation schemes on different templates, and then the above WENO reconstruction process is applied to the numerical viscosity term of entropy compatible schemes. The high precision entropy compatible scheme is obtained by combining with the high order entropy conserved scheme, and the high precision characteristic of the scheme is verified by several numerical examples. Moreover, the method of constructing weights by pressure and entropy is more time-saving than that by using characteristic variables directly.) the entropy compatible scheme and the high-resolution entropy compatible scheme are extended to two-dimensional cases. Several numerical examples of two-dimensional Euler equation are used to verify the characteristics of the scheme in two-dimensional case. The flow around a cylinder is mainly studied. The numerical examples show that the entropy compatible scheme can avoid the flow around a high Mach cylinder better than the Roe scheme. The erythema phenomenon in the problem.
【学位授予单位】：长安大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O241.82

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